Good reasoning. I should have also added that on these tests there is always an option E. NOTA (None of the above).

Actually a simple dissection of the triangle into 9 congruent triangles all similar to
the big triangle will show that the centre of mass divides the axis of symmetry
in the ratio of 2:1, again showing that (c) is the answer.

This is an elementary way to do this problem but it works here. No need to use the formula for finding a lamina just use a compass and straightedge. But that I mean the centroid of this isoseles triangle can easiliy be found by algebra.

Actualy a simple disection of the triangle into congruent triangles all similar to
the big triangle will show that the centre of mass divides the axis of symetry
in the ratio of 2:1, again showing that (c) is the answer.

Good reasoning. I should have also added that on these tests there is always an option E. NOTA (None of the above).

Another reason for choosing (c) now that I think of it is that the centre
of mass of a triangle is the point of intersection of the medians, and that
the medians divide one another into segments in the ratio 2:1.